# Tangential Acceleration Formula

View Notes

## Define Tangential Acceleration

For an object exhibiting a circular motion, there are always some parameters to describe its nature.

If we talk about a particle’s velocity, which is an angular velocity, that remains constant throughout the motion; however, angular acceleration makes two types of components and they are tangential and radial acceleration.

Tangential acceleration acts tangentially to the direction of motion of a particle and remains perpendicular to the direction of the radial component. Now, we’ll discuss the tangential and radiation acceleration formula.

### Tangential Acceleration and Centripetal Acceleration Formula

Tangential acceleration meaning is a measure of how the tangential velocity of a point at a given radius varies with time. Tangential acceleration is just like linear acceleration; however, it’s more inclined to the tangential direction, which is obviously related to circular motion.

If we talk about the narrow gap between the centripetal acceleration, which is an acceleration that acts towards the centre of the circle along which the body or a particle is creating a circular motion.

We can see there is a narrow line of difference between the two types of acceleration, and that the difference lies in the way the acceleration acts on the particle in a circular motion.

Now, let’s discuss the tangential acceleration equation followed by the centripetal acceleration.

### Tangential Acceleration Formula

Let’s suppose that you and your friends are playing with a string. You are in the middle of the string and your friends have joined the string from hand-to-hand and moving with high-speed or changing speed in a circular motion.

Here, we are talking about angular velocity and we know that change in the velocity is called acceleration, which is angular acceleration. So, we can write the first derivative of angular velocity with respect to time for angular acceleration.

### Acceleration Tangential Formula

Here, we aim to describe the tangential acceleration formula, so we will focus more on it, as our article relies on the same.

Now, writing the tangential acceleration equation in the

following manner:

Tangential acceleration (at)   = r x $\frac{d \omega}{dt}$ ….(1)

So, we denote the tangential acceleration with a subscript ‘ct’ along with the English letter ‘a’.

Here, $\frac{d \omega}{dt}$ =  angular acceleration

r = is the radius of the circle

Since the motion talks about the position of a particular object that’s why call ‘r’ as the radius vector.

We also know that the angular velocity can be written , so we can rewrite the above equation (1) to get the Tangential Acceleration Formula Circular Motion in a new form:

at = r𝛼….(2)

### Centripetal Acceleration Formula

The centripetal acceleration of an object making a circular motion with a circle ‘r’ and having a speed ‘v’ in meter per second is given by the following centripetal acceleration equation:

aC = v2/r

So, we denote the centripetal acceleration with a subscript ‘c’ along with the English letter ‘a’.

Now, we will discuss the radial and tangential acceleration formula in detail.

### Tangential and Radial Acceleration Formula

We already discussed the acceleration tangential formula in the above context, while talking about the narrow difference between the centripetal and tangential acceleration, we also saw a minor difference between tangential acceleration and the centripetal acceleration formula.

Now, let’s discuss the radial acceleration:

We define radial acceleration as the component that points along the radius vector, the position vector that points from a centre, usually of rotation, and the position of the particle that is accelerating.

The formula for radial acceleration is given by:

ar = v2/r …..(3)

Here, we can see the term ‘r’ or the radius vector has a difference in the tangential acceleration and the centripetal acceleration formula. Also, we notice that the centripetal acceleration and the radial acceleration have the same formula.

### Finding Tangential Acceleration

Now, we will look at one problem to find the tangential acceleration of an object.

Example:

If an object is experiencing a circular motion, then what will be its tangential acceleration? Also, determine the overall acceleration of the object.

The overall acceleration of an object is given by the following equation:

$\vec{a}_{(total)}$ = $\vec{a}_{r}$ + $\vec{a}_{t}$

Now, tangential acceleration can be determined by subtracting the radial component acceleration from the overall acceleration in the following manner:

$\vec{a}_{t}$  = $\vec{a}_{(total)}$ - $\vec{a}_{r}$

at θ(cap)  = $\vec{a}_{(total)}$ - $\vec{a}_{r}$ r (cap)

If we wish to find out the total acceleration in the modulus function, we have the following equation:

$\vec{a}_{(total)}$ = | $\vec{a}_{(total)}$ | = $\sqrt{a_{r}^{2} + a_{t}^{2}}$

So, the total acceleration is the square root of the sum of the squares of the radial and tangential acceleration.

1. Why do we study the rotational motion and what does the centripetal acceleration specify?

Rotational mechanics is one of the important topics of mechanics that requires great imagination and intuitive power. It helps us understand the mechanics behind the rotatory motion that we study in electric motors and generators.

In rotational motion, tangential acceleration is a measure of how fast a tangential velocity changes. It always acts orthogonally to the centripetal acceleration of a rotating object. It is equal to the product of angular acceleration α to the radius of the rotation. The tangential acceleration = radius of the rotation *  its angular acceleration.

It is always measured in radian per second square. Its dimensional formula is [T-2].

2. Why do we study tangential acceleration? What is a tangential velocity vector?

When an object makes a circular motion, it experiences both tangential and centripetal acceleration. Components of acceleration for a curved motion are radial and tangential acceleration.

The tangential component occurs because of the change in the speed of traversal. It points along the curve in the direction of the velocity vector; also in the opposite direction.

A tangential velocity works in the direction of a tangent at the point of circular motion. Henceforth, it always acts in the perpendicular direction to the centripetal acceleration of a rotating object. It always equals the product of angular acceleration with the radius of the rotation.